Let $ABC$ be a triangle and $A_1$, $B_1$, $C_1$ the midpoints of the sides $BC$, $CA$ and $AB$ respectively. Prove that if $M$ is a point in the plane of the triangle such that \[ \frac{MA}{MA_1} = \frac{MB}{MB_1} = \frac{MC}{MC_1} = 2 , \] then $M$ is the centroid of the triangle.

Let $M$ be the intersection of two diagonal, $AC$ and $BD$, of a rhombus $ABCD$, where angle $A<90^\circ$. Construct $O$ on segment $MC$ so that $OB<OC$ and let $t=\frac{MA}{MO}$, provided that $O \neq M$. Construct a circle that has $O$ as centre and goes through $B$ and $D$. Let the intersections between the circle and $AB$ be $B$ and $X$. Let the intersections between the circle and $BC$ be $B$ and $Y$. Let the intersections of $AC$ with $DX$ and $DY$ be $P$ and $Q$, respectively. Express $\frac{OQ}{OP}$ in terms of $t$.

3. Show that there are infinitely many triples (x,y,z) of integers such that $x^3 + y^4 = z^{31}$.

In $\triangle ABC$, let the bisector of $\angle BAC$ hit the circumcircle at $M$. Let $P$ be the intersection of $CM$ and $AB$. Denote by $(V,WX,YZ)$ the intersection of the line passing $V$ perpendicular to $WX$ with the line $YZ$. Prove that the points $(P,AM,AC), (P,AC,AM), (P,BC,MB)$ are collinear. [hide=Restatement]In isosceles triangle $APX$ with $AP=AX$, select a point $M$ on the altitude. $PM$ intersects $AX$ at $C$. The circumcircle of $ACM$ intersects $AP$ at $B$. A line passing through $P$ perpendicular to $BC$ intersects $MB$ at $Z$. Show that $XZ$ is perpendicular to $AP$.[/hide]

Consider an integer $n \geq 1$, $a_1,a_2, \ldots , a_n$ real numbers in $[-1,1]$ satisfying \begin{align*}a_1+a_2+\ldots +a_n=0 \end{align*} and a function $f: [-1,1] \mapsto \mathbb{R}$ such \begin{align*} \mid f(x)-f(y) \mid \le \mid x-y \mid \end{align*} for every $x,y \in [-1,1]$. Prove \begin{align*} \left| f(x) - \frac{f(a_1) +f(a_2) + \ldots + f(a_n)}{n} \right| \le 1 \end{align*} for every $x$ $\in [-1,1]$. For a given sequence $a_1,a_2, \ldots ,a_n$, Find $f$ and $x$ so hat the equality holds.

Let $n$ a natural number and $A=\{1, 2, 3, \cdots, 2^{n+1}-1\}$. Prove that if we choose $2n+1$ elements differents of the set $A$, then among them are three distinct number $a,b$ and $c$ such that $$bc<2a^2<4bc$$

We write in order of increasing number of 1 and all positive integers,which the sum of digits is divisible by $5$. Obtain a sequence of $1, 5, 14, 19. . .$ Prove that the n-th term of the sequence is less than $5n$.

Given $a$ and $b$ distinct positive integers, show that the system of equations $x y +zw = a$ $xz + yw = b$ has only finitely many solutions in integers $x, y, z,w$.

Find the sum of the prime factors of $67208001$, given that $23$ is one. [i]Proposed by Justin Stevens[/i]

Kevin is at the point $(19,12)$. He wants to walk to a point on the ellipse $9x^2 + 25y^2 = 8100$, and then walk to $(-24, 0)$. Find the shortest length that he has to walk. [i]Proposed by Kevin Zhao[/i]

Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$? [asy] unitsize(8); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((9,0)--(10,0)--(10,3)--(9,3)--cycle); draw((8,1)--(11,1)--(11,2)--(8,2)--cycle); draw((19,0)--(20,0)--(20,5)--(19,5)--cycle); draw((18,1)--(21,1)--(21,4)--(18,4)--cycle); draw((17,2)--(22,2)--(22,3)--(17,3)--cycle); draw((32,0)--(33,0)--(33,7)--(32,7)--cycle); draw((29,3)--(36,3)--(36,4)--(29,4)--cycle); draw((31,1)--(34,1)--(34,6)--(31,6)--cycle); draw((30,2)--(35,2)--(35,5)--(30,5)--cycle); label("Figure",(0.5,-1),S); label("$0$",(0.5,-2.5),S); label("Figure",(9.5,-1),S); label("$1$",(9.5,-2.5),S); label("Figure",(19.5,-1),S); label("$2$",(19.5,-2.5),S); label("Figure",(32.5,-1),S); label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$

When making a batch of $N \ge 5$ coins, a worker mistakenly made two coins from a different material (all coins look the same). The boss knows that there are exactly two such coins, that they weigh the same, but differ in weight from the others. The employee knows what coins these are and that they are lighter than others. He needs, after carrying out two weighings on cup scales without weights, to convince his boss that the coins are counterfeit easier than real ones, and in which coins are counterfeit. Can he do it?

(B.Frenkin, 8) Does a regular polygon exist such that just half of its diagonals are parallel to its sides?

We have two concentric circles. A polygon is circumscribed around the smaller circle and is contained entirely inside the greater circle. Perpendiculars from the common center of the circles to the sides of the polygon are extended till they intersect the greater circle. Each of the points obtained is connected with the endpoints of the corresponding side of the polygon . When is the resulting star-shaped polygon the unfolding of a pyramid?

The cells of an $n\times n$ board with $n\ge 5$ are coloured black or white so that no three adjacent squares in a row, column or diagonal are the same colour. Show that for any $3\times 3$ square within the board, two of its corner squares are coloured black and two are coloured white.

Two circles with radii $R$ and $r$ intersect at $C$ and $D$ and are tangent to a line $\ell$ at $A$ and $B$. Prove that the circumradius of triangle $ABC$ does not depend on the length of segment $AB$.

Given an integer $n>2$ and an integer $a$, if there exists an integer $d$ such that $n\mid a^d-1$ and $n\nmid a^{d-1}+\cdots+1$, we say [i]$a$ is $n-$separating[/i]. Given any n>2, let the [i]defect of $n$[/i] be defined as the number of integers $a$ such that $0<a<n$, $(a,n)=1$, and $a$ is not [i] $n-$separating[/i]. Determine all integers $n>2$ whose defect is equal to the smallest possible value.

A $1 \times 5n$ rectangle is partitioned into tiles, each of the tile being either a separate $1 \times 1$ square or a broken domino consisting of two such squares separated by four squares (not belonging to the domino). Prove that the number of such partitions is a perfect fifth power. [i](K. Kokhas)[/i]

Find all integers $k\ge 2$ such that for all integers $n\ge 2$, $n$ does not divide the greatest odd divisor of $k^n+1$.

Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20. [i]Proposed by Luxembourg[/i]

The set $M= \{1;2;3;\ldots ; 29;30\}$ is divided in $k$ subsets such that if $a+b=n^2, (a,b \in M, a\neq b, n$ is an integer number $)$, then $a$ and $b$ belong different subsets. Determine the minimum value of $k$.

For a permutation $\pi$ of the set $A = \{1, 2, \ldots, 2025\}$, define its [i]colorfulness [/i]as the greatest natural number $k$ such that: - For all $1 \le i, j \le 2025$, $i \ne j$, if $|i - j| < k$, then $|\pi(i) - \pi(j)| \ge k$. What is the maximum possible colorfulness of a permutation of the set $A$? Determine how many such permutations have maximal colorfulness. [i]Proposed by Pavle Martinović[/i]

Two circles $C_{1}$ and $C_{2}$ intersect in $A$ and $B$. A line passing through $B$ intersects $C_{1}$ in $C$ and $C_{2}$ in $D$. Another line passing through $B$ intersects $C_{1}$ in $E$ and $C_{2}$ in $F$, $(CF)$ intersects $C_{1}$ and $C_{2}$ in $P$ and $Q$ respectively. Make sure that in your diagram, $B, E, C, A, P \in C_{1}$ and $B, D, F, A, Q \in C_{2}$ in this order. Let $M$ and $N$ be the middles of the arcs $BP$ and $BQ$ respectively. Prove that if $CD=EF$, then the points $C,F,M,N$ are cocylic in this order.

Let $f(x)$ be a function satisfying $f(x+1)-f(x)=2x+1 (x \in \mathbb{R})$.In addition, $|f(x)|\le 1$ holds for $x\in [0,1]$. Prove that $|f(x)|\le 2+x^2$ holds for $x \in \mathbb{R}$.

Let $x,y,z$ be complex numbers satisfying \begin{align*} z^2 + 5x &= 10z \\ y^2 + 5z &= 10y \\ x^2 + 5y &= 10x \end{align*} Find the sum of all possible values of $z$. [i]Proposed by Aaron Lin[/i]